We discuss the cobordism type of spin manifolds with nonnegative sectional curvature. We show that in each dimension $4k geq 12$, there are infinitely many cobordism types of simply connected and nonnegatively curved spin manifolds. Moreover, we rais
e and analyze a question about possible cobordism obstructions to nonnegative curvature.
We give new counterexamples to a question of Karsten Grove, whether there are only finitely many rational homotopy types among simply connected manifolds satisfying the assumptions of Gromovs Betti number theorem. Our counterexamples are homogeneous
Riemannian manifolds, in contrast to previous ones. They consist of two families in dimensions 13 and 22. Both families are nonnegatively curved with an additional upper curvature bound and differ already by the ring structure of their cohomology rings with complex coefficients. The 22-dimensional examples also admit almost nonnegative curvature operator with respect to homogeneous metrics.
We give a characterization of closed, simply connected, rationally elliptic 6-manifolds in terms of their rational cohomology rings and a partial classification of their real cohomology rings. We classify rational, real and complex homotopy types of
closed, simply connected, rationally elliptic 7-manifolds. We give partial results in dimensions 8 and 9.
We study manifolds with almost nonnegative curvature operator (ANCO) and provide first examples of closed simply connected ANCO mannifolds that do not admit nonnegative curvature operator.